Totally nondegenerate surfaces in CR dimension one are maximally homogeneous and standard.
problem Understanding the properties of totally nondegenerate surfaces in CR dimension one.
method Analyzing Levi-Tanaka and infinitesimal CR automorphism algebras.
result Totally nondegenerate surfaces in CR dimension one are maximally homogeneous and standard.
Study on CR structures in 7D, proving maximal symmetry dimension.
problem Proving symmetry dimension bound for CR structures in 7D.
method Investigating homogeneous models and proving uniqueness.
result 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in 7D.
Paper studies CR submanifolds in complex projective space with flat normal connection.
problem Existence and properties of CR submanifolds with flat normal connection.
method Investigation of umbilical normal vector and application to non-existence proof.
result Non-existence of certain CR submanifolds of maximal CR dimension.
Classifies maximal symmetry models of CR dimension 1.
problem Classifying maximal symmetry models of CR manifolds.
method Classification based on Bloom-Graham and Tanaka types, coordinate realization, and extension principle.
result Classification and coordinate realization of maximal symmetry models.
New submaximal symmetry dimensions found for CR-structures.
problem Determining the maximum symmetry for CR-structures.
method Analyzing hypersurface type CR-structures with non-degenerate Levi forms.
result Submaximal symmetry dimensions for Levi-indefinite and Levi-definite structures.
It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.
Characterizes chains in 3D CR and para-CR structures.
problem Determining when a 3D path geometry comes from CR or para-CR chains.
method Provides necessary and sufficient conditions for a 3D path geometry to arise from chains of CR or para-CR 3-manifolds, and verifies computationally.
result Characterization of chains in 3D CR and para-CR structures.
Analyzes the Levi form on CR manifolds of any dimension.
problem Understanding the Levi form on CR manifolds of varying dimensions and codimensions.
method Analytical and geometrical study of the Levi form.
result Comprehensive insights into the Levi form on CR manifolds.
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.
Study shows CR models' rigidity, proving Beloshapka's conjecture.
problem CR model rigidity and Beloshapka's conjecture.
method Elie Cartan's classical method and weight analysis of structure equations.
result CR automorphism Lie groups do not contain nonlinear maps.
CR Yamabe flow converges on specific CR manifolds.
problem Convergence of CR Yamabe flow on compact CR manifolds.
method Proving convergence for n=1 or spherical manifolds. result CR Yamabe flow converges under specific conditions.
Paper studies CR holomorphic functions in Sasakian manifolds.
problem Sharp dimension estimate of CR holomorphic functions in Sasakian manifolds.
method Focuses on CR analogue of Yau's uniformization conjecture in Sasakian manifolds.
result Establishes the sharp dimension estimate of CR holomorphic functions.
Paper shows CR Q-curvature orthogonal to CR pluriharmonic functions.
problem Understanding CR Q-curvature and pluriharmonic functions on CR manifolds. method Obtained a cohomological expression for the integral of CR Q-curvature and pluriharmonic functions. result CR Q-curvature is orthogonal to CR pluriharmonic functions. Investigates CR structures in 7D, showing 8 is max symmetry dimension.
problem CR structures in 7D with intransitive symmetry.
method Various methods to bound symmetry dimension, demonstrating existence of non-equivalent models.
result Existence of infinitely many non-equivalent submaximally symmetric models.
Let M be a CR submanifold of maximal CR dimension of a complex space form M. The shape operator A of the distinguished vector field ξ is recurrent if there exists a 1-form v such that \nabla A = A \otimes v. We show that M is an Euclidean space under the condition that A is recurrent.
Defines Lewy curves in para-CR geometry and characterizes their path geometries.
problem Characterizing path geometries defined by para-CR Lewy curves.
method Definition and characterization of para-CR Lewy curves in various dimensions.
result Lewy curves determine the para-CR structure up to sign in flat cases.
Researchers found new dimensions for exceptional Lie group realizations.
problem Determining the lowest dimensions for exceptional Lie group realizations.
method Realized CR structures in dimensions 16 and 24 for E6 and E8. result Discovered new realizations of E6 and E8 in dimensions 16 and 24. We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded En…
Study on Hausdorff dimension of singular CR Yamabe problem.
problem Estimating the Hausdorff dimension of the singular set in CR geometry.
method Conformal deformation of contact form to solve Yamabe problem and estimate Hausdorff dimension.
result Estimates the Hausdorff dimension of the singular set in CR geometry.
It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …
The paper calculates variations of Einstein-Hilbert action on CR manifolds.
problem Variation of the Einstein-Hilbert action in pseudohermitian geometry.
method Computed first and second variations on CR manifolds, characterized critical points as pseudo-Einstein structures, and analyzed second variation on standard spheres.
result In three dimensions, the second variation of the Einstein-Hilbert action on CR structures differs from the Riemannian case due to embeddability.
The Fefferman metric connects CR manifolds to conformal geodesics in 3D.
problem Understanding the Fefferman metric on CR manifolds.
method Explicit description of the Fefferman metric and variational characterization of conformal geodesics.
result Conformal geodesics have lifts to chains and null chains, and are characterized by total torsion.
Develops new approach to recover CR structures from their Levi foliations.
problem Recovering CR structures from their Levi foliations for nonregular symbols.
method Reduction to dynamical Legendrian contact structure on leaf space.
result New geometric interpretation of CR prolongation conditions.
Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and suffcient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that the normal almost paracontact …
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
Defines and proves CR invariants on five-manifolds.
problem Defines and studies CR invariants on CR five-manifolds.
method Defines global secondary CR invariants and proves their linear combination.
result Any global secondary CR invariant is a linear combination of total Q′-curvature, total I′-curvature, and a local CR invariant. It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.
The paper explores unique properties of Kähler manifolds without shared CR-submanifolds.
problem Characterizing Kähler manifolds with specific CR-submanifolds.
method Analyzing properties of Kähler and CR-submanifolds in complex manifolds.
result Kähler manifolds without shared CR-submanifolds have distinct properties.
For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kern…
Paper finds fractional Q-curvature on 3D CR sphere exists.
problem Existence of fractional Q-curvature on 3D CR sphere.
method Critical points theory at infinity.
result Existence result for fractional Q-curvature problem.
The paper identifies all flat CR Lie groups and their structures.
problem Identifying flat CR Lie groups and their structures.
method Analyzing Lie algebras and structures of Lie groups.
result Only specific Lie algebras are Cartan flat non-degenerate CR Lie algebras.
Researchers create higher-dimensional I′-curvatures and find counterexamples to the Hirachi conjecture.
problem The Hirachi conjecture in higher CR dimensions.
method Constructing higher-dimensional I′-curvatures and analyzing their properties under contact form changes. result Total integrals of I′-curvatures depend on the choice of contact form, providing counterexamples to the Hirachi conjecture. Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.
problem Classifying CR hypersurfaces with maximal symmetry in low dimensions.
method Introduced modified CR symbols to organize local invariants, classified hypersurfaces through modified symbols, and used Lie group structures.
result Found nine model structures among locally homogeneous 2-nondegenerate hypersurfaces in C4. We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding …
The paper sets a limit on CR manifold properties.
problem Understanding properties of CR manifolds.
method Analyzes Chern classes of specific manifolds.
result Establishes an optimal constraint on CR manifolds.
Constructs Einstein ACH metrics with CR structures, proving CR GJMS operators exist.
problem Constructing Einstein ACH metrics with specified CR structures.
method Refined Matsumoto's construction, solving Einstein equation to infinite order.
result Self-dual Einstein ACH metrics constructed for CR structures.
Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. New CR manifolds found with same Kohn Laplacian spectra.
problem Spectrum of Kohn Laplacian doesn't determine CR manifold equivalence.
method Constructed pairs of odd-dimensional elliptic manifolds.
result Found manifolds with same spectrum but different CR structures.
Study reveals CR structure of snake robot's geometry.
problem Understanding the kinematics and geometry of a snake robot.
method Analysis of (2,3,5) distributions and solving Cartan equivalence problem.
result Discovery of a CR structure with CR dimension 1 and real codimension 3.
We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontr…
This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…
The paper constructs global CR invariants from renormalized characteristic forms.
problem Global CR invariants on strictly pseudoconvex domains.
method Renormalized characteristic forms of the Cheng--Yau metric.
result Generalizations of I′-curvature on CR five-manifolds. We study the equivalence problem for 4-dimensional CR-manifolds of CR-dimension 1 and codimension 2 which are referred to as Engel CR-manifolds. We construct a canonical Cartan connection on such CR-manifolds through Cartan equivalence's method. In particular, we give the explicit expression of 4 biholomorphic …
We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of the 3-sphere or of a circle bundle over a Riemann surface of positive genus.…
In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface $M\subset \bC^{n+1}$ into a sphere $\bS$ in a higher dimensional complex space $\bC^{N+1}$. The degeneracy of the mapping f will be characterized in terms of the ranks of the CR second fundamental form and its covariant deriv…
3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Paper proves existence of minimum energy solutions in 5D contact spin manifolds.
problem Finding minimum energy solutions for CR Yamabe equation in 5D contact spin manifolds.
method Spinorial approach based on a positive mass theorem.
result Existence of minimum energy solutions in 5D contact spin manifolds.